Falling stars: a fall-decorated rational shuffle theorem

Abstract In this paper, we formulate a rational analog of the fall Delta theorem and the Delta square conjecture. We find a new dinv statistic on fall-decorated paths on a ( m + k ) × ( n + k ) $(m+k) \times (n+k)$ left parenthesis m plus k right parenthesis times left parenthesis n plus k right parenthesis rectangle that simultaneously extends the previously known dinv statistics on decorated square objects and nondecorated rectangular objects. We prove a symmetric function formula for the q , t $q,t$ q comma t -generating function of fall-decorated rectangular Dyck paths as a skewing operator applied to e m , n + k m $e_{m,n+km}$ e Subscript m comma n plus k m and, conditionally on the rectangular paths conjecture, an analog formula for fall-decorated rectangular paths.